Objective and Subjective Specifications

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1 Objective and Subjective Specifications Eric C.R. Hehner Department of Computer Science, University of Toronto hehner@cs.utoronto.ca Abstract: We examine specifications for dependence on the agent that performs them. We look at the consequences for the Church-Turing Thesis and for the Halting Problem. Introduction The specifications considered in this paper are specifications of behavior, or activity. I include human behavior, computer behavior, and other behavior. To keep the examples simple, I will use specifications that say what the output, or final state, of the behavior should be. And I will use specifications that relate input, or initial state, to output, or final state. The conclusions apply also to specifications that say what the interactions during the behavior should be, but my examples will not be that complicated. A specification may have the form of a question, for example What is two plus two?. Or it may have the form of a command, for example Tell me what is two plus two.. The question and the command are equivalent because they invoke the same behavior. A specification may describe the desired behavior, for example, saying what is two plus two. Definitions A specification is objective if the specified behavior does not vary depending on the agent that performs it. For examples: (0) Given a natural number, what is its square? (1) What is the number of words in this question? (2) What is the name of the first Turing Award winner? For all three questions, the correct answer does not depend on who or what is answering. A specification is subjective if the specified behavior varies depending on the agent that performs it. For examples: (3) What is your name? (4) What is your IP address? The correct answer to question (3) depends on whom you ask. In this paper, subjective does not mean that the answer is a matter of disagreement, debate, doubt, or dishonesty. If we ask Alice what her name is, the answer Alice is correct, and all other answers are wrong. If we ask Bob the same question, the correct answer is different. Question (4) is similar to question (3), but applies to a computer rather than a human. Subjectively and Objectively Inconsistent Now consider this example: (5) Lift Bob. I am not interested in the variety of lifting techniques; I am interested only in the specified result: the agent lifts Bob. If we ask Hercules, who is very strong, to lift Bob, he can do so without difficulty. If we ask Alice, who is much smaller than Bob, she is not strong enough. The result is different, depending on who is trying to lift Bob. So it may seem that (5) is subjective. But the definition of subjective specification says the specified behavior varies depending on the agent. When we ask Alice to lift Bob, we are asking for the same behavior

2 1 Eric Hehner (lifting Bob) as when we ask Hercules. So it may seem that (5) is objective. But suppose we ask Bob to lift Bob. He cannot do so, but not due to lack of strength. He cannot do so because the specification does not make sense when we ask Bob to lift himself. The specification makes sense for some agent (anyone other than Bob), and makes no sense for some agent (Bob). For that reason, (5) is subjective. If we restrict the set of agents to exclude Bob, then (5) is objective. (6) Can Carol correctly answer no to this question? Let's ask Carol. If she says yes, she's saying that no is the correct answer for her, so yes is incorrect. If she says no, she's saying that she cannot correctly answer no, which is her answer. So both answers are incorrect. Carol cannot answer the question correctly. Now let's ask Dave. He says no, and he is correct because Carol cannot correctly answer no. So (6) is subjective because it is a consistent, satisfiable specification for some agent (anyone other than Carol), and an inconsistent, unsatisfiable specification for some agent (Carol). (7) Can any man correctly answer no to this question? Let's ask Ed, who is a man. Suppose Ed says no. Ed is saying that no man can correctly answer no, and Ed, a man, is answering no, so Ed is saying that his answer is incorrect. Suppose Ed says yes. Ed is saying that some man, let's call him Frank, can correctly answer no. But if Frank answers no, he is saying that his own answer is incorrect. So Frank cannot say no correctly. So Ed's yes answer is incorrect. And the same goes for every man. But Gloria, who is not a man, can correctly say no. Specification (7) is subjective because it is a consistent, satisfiable specification for some agent (anyone who is not a man), and an inconsistent, unsatisfiable specification for some agent (any man). (8) Can anyone correctly answer no to this question? If we ask Harry and he says no, he is saying that his answer is incorrect. If he says yes, he is saying that someone, let's say Irene, can correctly answer no. But if Irene answers no, she is saying that her answer is incorrect. So Harry can neither say no nor yes correctly. And the same goes for anyone else we ask. The correct answer to the question is therefore no, but no-one can correctly say so (oops, I just did). I meant: no-one who is a possible agent can say so. I exclude myself from the set of possible agents just so that I can tell you that no possible agent can correctly answer no. Specification (8) is objectively inconsistent. Specifications (6), (7), and (8) are examples of twisted self-reference. The self-reference occurs when the specification talks about the agent who will perform the specification. The twist, in these examples, is the word no. If we replace no with yes in these three specifications, then everyone can correctly answer yes to all of them, making them objectively consistent. Church-Turing Thesis If a specification can be computed by any one of: a Turing Machine (a kind of computer) [2] the lambda-calculus (a mathematical formalism) [0] general recursive functions (another mathematical formalism) [1] then it can be computed by all of them; they all have the same computing power. The Church- Turing Thesis [3] says that each of these formalisms compute all that is computable. In a more modern version, the Church-Turing Thesis says that if a specification can be computed, then it can be computed by a program in any general-purpose programming language. All generalpurpose programming languages provide the same computing power; they are Turing-Machine- Equivalent (TME).

3 Objective and Subjective Specifications 2 Church and Turing were thinking of specifications of mathematical functions, like (0). It seems reasonable to me that the Church-Turing Thesis can be extended to all objective specifications. But its extension to subjective specifications comes up against a problem. Reconsider subjective specification (7), but replace man with L-program, meaning a program written in TME-language L. (L can be Pascal, or Python, or any other programming language.) (9) Can any L-program correctly answer no to this question? It's easy to write an L-program that prints no. If that is the answer to (9), it is saying that there isn't an L-program that correctly answers no to the question, so in particular, the L- program that prints no doesn't give the correct answer. It's just as easy to write an L-program that prints yes. If that program is the answer to (9), it says that no is the correct answer, so the L-program that prints yes doesn't give the correct answer either. In fact, the correct answer to (9) is no, but no L-program can correctly say so. We can write a program in language M (which is another TME-language) that prints no in answer to (9), and that answer is correct. No matter whether the agents are people or programs, the result is the same: one agent can satisfy the specification, but another can't. The Church-Turing Thesis, in the version stated earlier, does not apply to subjective specifications. Specification (9) can be computed by a program in TME-language M, but not by a program in TME-language L. Another version of the Church-Turing Thesis is that any program in any TME-language can be translated to a program in any other TME-language. We'll look at program translation in a moment, but first, here is a non-programming example to illustrate the translation problem. (10) Is this question in French? The correct answer is no. The question is easily translated into French. (11) Cette question est-elle en français? The correct answer is now oui. Before translation, when the question is put to someone who understands the language the question is in, it invokes one behavior: saying no. After an accurate translation, when the question is put to someone who understands the language the question is now in, it invokes a different behavior: saying oui. Specifications (10) and (11) refer to a language, and changing the language of the question affects the answer. Similarly, when we write a program to compute a subjective specification, then translate it to another language, it may invoke different behavior. This can occur when the specification refers to a programming language. First, here's an objective specification that refers to a programming language. (12) Is text p an L-program? Every compiler answers the question whether its input text is a program in the language that it compiles. Whether we write the program that computes (12) in language L or in language M, for the same input p we should get the same answer. Specification (12) is objective, and the Church-Turing Thesis applies. Now replace the input with a self-reference. (13) Is the program answering this question an L-program? There are two ways to write an M-program to compute (13). The hard way is to give the program access to its own text, perform the lexical analysis and parsing and type checking and so on, just as a compiler would do, and then print the answer, which is no. The easy way is just to print no because that's the right answer. Now we translate our M-program to language L. If we programmed the hard way, the translated program accesses its own text, does the analysis, and prints the correct answer, which is yes. If we programmed the easy way, the translated program prints no, which is incorrect. Specification (13) is subjective, and the Church-Turing Thesis does not apply. Either the translation prints the correct answer by exhibiting different behavior, or the translation exhibits the same behavior and the answer is incorrect.

4 3 Eric Hehner The same choice, whether to preserve the behavior or to preserve the specification, can occur without translation, simply by renaming. For example, (14) Given a text p representing a program, determine whether p calls the determining program. Let's name the determining program DoYouCallMe. Given program p, it searches within p for a calling occurrence of DoYouCallMe, reporting yes if one is found, and no if not. Now let's rename the determining program DoYouCallMe2. If we just change the name of the program without changing what it is searching for, this name change preserves behavior, but the program no longer satisfies the specification (14). If we change both the program name and what it is searching for, the program still satisfies the specification (14), but its behavior changes: given the same input, DoYouCallMe and DoYouCallMe2 may give different answers. If a program has access to its own name, then changing its name automatically changes what it is searching for; the result still satisfies (14), but has different behavior. Yet another version of the Church-Turing Thesis is that in any TME-language, you can write an interpreter for programs in any other TME-language. Interpretation is the same as executing a translation, and it is similarly limited to programs that satisfy objective specifications. Given a program that computes a subjective specification, its interpretation may produce behavior that differs from execution of the given program. Halting Problem, Language-Based When Alan Turing laid the foundation for computation in 1936 [2], he wanted to show what computation can do, and what it cannot do. For the latter, he invented a problem that we now call the Halting Problem. Without loss of generality and without changing the character of the problem, I consider halting for programs with no input (input to a program can always be replaced by a definition within the program). In modern terms, it is as follows. (15) Given a text p representing an L-program that requires no input, report stops if execution of p terminates, and loops if execution of p does not terminate. The input p represents a program in TME-language L. The agent that performs specification (15) must be a program, written in a TME-language, running on a computer. (In fact, Turing used the word machine for the combination of program and computer.) I am excluding distributed computations so that I can identify the agent. First, let's ask for a program written in language L to perform (15), and let's call it halts. If there is such a program, then there is also a program in language L, let's call it diag, whose execution is as follows: diag calls halts ( diag ) to determine if its own ( diag 's) execution will terminate. If halts reports that diag 's execution will terminate, then diag 's execution becomes a nonterminating loop; otherwise diag 's execution terminates. We assume there is a dictionary of function and procedure definitions that is accessible to halts, so that the call halts ( diag ) allows halts to look up diag and halts in the dictionary, and retrieve their texts for analysis. When programmed in language L, specification (15) is another twisted self-reference. The selfreference is indirect: halts applies to diag, and diag calls halts. The twist is supplied by diag. If halts reports that diag 's execution will terminate, then diag 's execution is a nonterminating loop. If halts reports that diag 's execution will not terminate, then diag 's execution terminates. Whatever halts reports about diag, it is wrong. Therefore specification (15) is inconsistent when we ask for a program written in language L to perform it.

5 Objective and Subjective Specifications 4 Now let's ask for a program written in TME-language M to perform (15), where M is such that programs written in L cannot call programs written in M. Can this M-program be written? Since L-programs cannot call M-programs, we cannot rule it out by a twisted self-reference. I present two possible answers to the question. Answer O: Specification (15) is objective, like specification (12). But unlike (12), it is an inconsistent specification, no matter what language we use. If we could write an M-program to compute halting for all L-programs, we could translate it into L (or interpret it by an L- program), and because (15) is objective, the translation (or interpretation) would also compute halting correctly for all L-programs. But there is no L-program to compute halting for all L- programs. So there is no program in any language to compute halting for all L-programs. Answer S: Specification (15) is subjective. Like specification (13), (15) refers to a programming language L. When programmed in L there is a twisted self-reference; when programmed in M there is no self-reference. There is an M-program to compute halting for all L-programs. Because (15) is subjective, its translation to L (or interpretation in L) does not compute halting for all L-programs. Perhaps the M-program says correctly that diag 's execution terminates, and its translation to L (or interpretation in L), which we call halts, says incorrectly that diag 's execution does not terminate, and that is why it terminates. Answer O has been almost unanimously accepted by computer scientists, but its acceptance is premature because (15) has never been shown to be objective, and Answer S has never been ruled out. I favor Answer S for the weak reason that I cannot see any inconsistency in asking for an M-program to compute halting for all L-programs. Halting Problem, Machine-Based The preceding discussion of halting is language-based. Here is a similar discussion that is computer-based. Suppose there are two identical disconnected computers C and D, and all programs are written in the same TME-language, and all programs can run on either computer. Both computers have enough memory so that memory limitation is not an issue. (16) Given a text p representing a program that requires no input, loaded on computer C, report stops if execution of p terminates, and loops if execution of p does not terminate. The agent that performs specification (16) must be a program running on either C or D. Once again, I exclude distributed computing so that I can identify the agent, and once again I assume there is a dictionary of function and procedure definitions on each computer. First, let's ask for a program running on computer C to perform (16), and let's call it halts. If there is such a program, then we can write another program, let's call it diag, exactly as before, and we can load this program onto computer C. As before, diag calls halts to report on diag, and then diag does the opposite; so whatever halts reports, it is wrong. Specification (16) is inconsistent when we ask for a program running on computer C to perform it. Now let's ask for a program running on computer D to perform (16). Can this program be written? Since programs on C cannot call programs on D, we cannot rule it out by a twisted self-reference. As in the language-based case, we have the same two possible answers to the question: Answer O and Answer S. Answer O: Specification (16) is objective. It is an inconsistent specification, no matter what computer we use. There is no program on any computer to compute halting for programs on computer C.

6 5 Eric Hehner Answer S: Specification (16) is subjective. There is a program on computer D, and again let's call it halts, to compute halting for all programs on computer C. We can carry the halts program from D to C and run it there. But when we run it on C, it does not compute halting for all programs on C. This is quite counter-intuitive. When halts applies to diag, and diag calls halts, it matters whether the halts that applies (the first occurrence of halts in this sentence) is the same instance as the halts that is called (the second occurrence of halts in this sentence). In one case, there is a twisted self-reference, and in the other, there isn't, and that can affect the computation. Conclusion A specification is objective if the specified behavior does not depend on the agent that performs it, and subjective if it does. The Church-Turing Thesis applies to objective specifications, not to subjective ones. The Halting Problem may be a subjective specification. It is inconsistent to ask for an X-program to compute halting for all X-programs, but it may be consistent and satisfiable to ask a Y-program to compute halting for all X-programs, where X and Y can be two programming languages, or two computers. At least it has not yet been proven impossible. Acknowledgement I thank Bill Stoddart for stimulating discussions. References [0] A.Church: the Calculi of Lambda Conversion, Princeton University Press, 1941 [1] K.Gödel, reported by S.C.Kleene: Introduction to Metamathematics, North-Holland, 1951 [2] A.M.Turing: on Computable Numbers with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society s.2 v.42 p , 1936; correction s.2 v.43 p , 1937 [3] A.M.Turing: Systems of Logic Based on Ordinals, p.8, Ph.D. thesis, Princeton University, 1939 other papers on halting

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2015-1-2 0 How to Compute Halting Eric C.R. Hehner Department of Computer Science, University of Toronto hehner@cs.utoronto.ca Abstract: A consistently specified halting function may be computed. Halting